Operator gradient of divergencie in subspaces of $\mathbf{L}_{2}(G)$ space

Abstract: The author studies the structure of space $ \mathbf {L} _ {2} (G) $ of vector-valued functions that are square integrable in a bounded connected domain $ G $ of the three-dimensional space with a smooth boundary and the role of gradient divergence operators and the rotor in the construction of bases in subspaces $ {\mathcal {{A}}} $ and $ {\mathcal {{B}}} $. The self-adjointness of the extension $ \mathcal {N} _d $ of operator $ \nabla \mathrm {div} $ to the subspace $ \mathcal {A} _ {\gamma} \subset {\mathcal {{A}}} $ and the basicity system of its own functions. Written explicit formulas for solving the spectral problem in a ball and the conditions for the decomposition vector-functions in a Fourier series in eigenfunctions gradient of divergence. The solvability of the boundary tasks: $ \nabla \mathrm {div} \, \mathbf {u} + \lambda \, \mathbf {u} = \mathbf {f} $ in $ G $, $ (\mathbf {n} \cdot \mathbf {u}) | _ {\Gamma} = g $ in Sobolev spaces $ \mathbf {H} ^ {s} (G) $ of order $ s \geq 0 $ and in subspaces. In passing, similar results for the operator of the rotor and its symmetric extension $ S $ to $ \mathcal {B} $.